Complex Analysis and Dynamics Seminar

Fall 2013 Schedule

We investigate the so-called persistence problem of Sloane, exploiting connections with the dynamics
of circle maps and the ergodic theory of $\mathbb{Z}^d$ actions. We also formulate a conjecture,
concerning the asymptotic distribution of digits in long products of primes chosen from a given finite set,
whose truth would in particular solve the persistence problem. We provide computational evidence and
a heuristic argument in favor of our conjecture. Such heuristics can be thought in terms of a simple model
in statistical mechanics. This talk is based on joint work with Charles Tresser (IBM).

Let $S_{g,p}$ be the orientable surface of genus $g$ with $p$ punctures, and let
$W(S)= 3g+p-4>0$. Given a natural number $k$, we demonstrate a lower bound
on the geometric intersection number for any pair of curves that are a distance of $k$
apart in the corresponding curve graph, and which grows "almost polynomially" of
degree $k-2$, in the sense that it grows faster than $(W(S)/2)^{c(k-2)}$, for any
$c \in (0,1)$. We use this to prove that curve graphs are uniformly hyperbolic, and
that train track splitting sequences project to $R$-quasigeodesics in the curve graph
of any essential subsurface, where $R= O(W(S)^2)$. Time permitting, we will demonstrate
a construction which shows that these lower bounds on intersection
numbers are asymptotically sharp in some suitable sense.

The set of closed hyperbolic surfaces of genus $g$ up to isometry (moduli space $M_g$)
admits a number of geometries of its own. Although there exists a rich theory describing
the geometry of moduli space, surprisingly little is understood about qualities that
might be described as aspects of its "shape" when equipped with a natural metric. An intriguing subspace
of $M_g$ is given by the subset consisting of surfaces whose systoles fill the surface (the systoles are shortest simple closed geodesics). In this talk I'll describe a first attempt at trying to understand the shape of surfaces with systoles who fill by comparing it to other natural subsets of $M_g$. This is joint work with J. Anderson and A. Pettet.

The thermodynamic formalism has played an important role in the
development of the statistical theory of dynamical systems. The main player
in this topic is the topological pressure that connects topological and measure-theoretic properties of the
dynamical systems. This connection is established by the so-called
variational principle.
In this talk we introduce a natural notion of localized topological
pressure and discuss several of its variational properties. We derive the local
variational principle for a wide variety of spaces and potentials but
also obtain counterexamples.
Next, we discuss localized equilibrium states and show that even in the
case of systems and potentials with strong thermodynamic properties the
classical theory of equilibrium states breaks down. The results presented
in this talk are joint work with Tamara Kucherenko.

In William Thurston's last paper, "Entropy in Dimension One," he completely characterizes which numbers arise as $\exp$(entropy($f$)), where $f$ is a critically finite real polynomial map of a closed interval. Inspired by his work (and the spectacular fractal picture on page 1 of his manuscript), we consider a particular dynamical quantity associated to critically finite rational maps. Following earlier work of Thurston, a critically finite rational map induces a holomorphic endomorphism on a Teichmueller space, and this endomorphism has a unique fixed point. We study the spectrum of the derivative of this endomorphism and prove that there is a prominent spectral gap in the case of quadratic polynomials. We plot a picture for this data (analogous to Thurston's entropy picture), revealing some incredible fractal structure. This is joint work with X. Buff and A. Epstein.

Nov 1: The seminar will feature two talks:

2:00-3:00 Arnaud Cheritat (University of Toulouse)
On the Size of Siegel Disks of Degree 3 Polynomials

For degree 2 polynomials, a necessary and sufficient condition for linearizability of indifferent periodic points is known:
it depends only on the eigenvalue of the fixed point, and is called Brjuno's condition. Decades ago Douady conjectured that
the condition is the same for any degree. A lot is known on the inner size of the Siegel disk in degree 2. Xavier Buff formulated
an interesting conjecture for higher degree. These problems seem beyond reach for now, so we study related questions.
Of particular interest is the way the size of a Siegel disk varies when the eigenvalue is fixed. I will explain some joint results
with Buff, and I will present new results by Ilies Zidane, and many conjectures.

3:30-4:30 Misha Yampolsky (University of Toronto)
Note: This talk will be in Room 5383Geometrization of Branched Coverings of the Sphere and Decidability of Thurston Equivalence

I will discuss a recent joint work with N. Selinger on constructive geometrization of branched coverings of the 2-sphere.
I will further describe the connection between geometrization and the general question of algorithmic decidability of Thurston
equivalence, and will present a new decidability result obtained jointly with Selinger, which generalizes my previous work with
M. Braverman and S. Bonnot.

Nov 8: Jun Hu (Brooklyn College and Graduate Center of CUNY) Douady-Earle Extensions: Further Investigations and New Applications

In this talk, I will give a very brief summary of the main results of four papers in which you find
either further investigations or new applications of Douady-Earle extensions of circle homeomorphisms:
(1) The Douady-Earle extension of a circle diffeomorphism extends to a diffeomorphism of the
sphere (joint work with Susovan Pal).
(2) How the maximal dilatation of the extension is universally controlled on a universal neighborhood of
the origin by distortions of the boundary map at finitely many points (joint work with Oleg Muzician).
(3) A metric characterization of the asymptotic Teichmuller space of the open unit disk through equivalence
classes of the shear functions of circle homeomorphisms (joint work with Jinhua Fan).
(4) The Kobayashi and Teichmuller metrics coincide on the Weil-Petersson and VMOA Teichmuller spaces
(joint work with Jinhua Fan).

Nov 15: Zhe Wang (Bronx Community College of CUNY)
Motions on the Riemann Sphere

In this talk, I will discuss part of a recent joint work with Yunping Jiang, Sudeb Mitra and Hiroshige Shiga on the extension problem of continuous motions, quasiconformal motions, local quasiconformal motions, and holomorphic motions. We give an example of a quasiconformal motion of an infinite set in the Riemann sphere, over an interval, that cannot be extended to a quasiconformal motion of the sphere. Following this example, we introduce a new concept called local quasiconformal motion. With this new definition, we show that any local quasiconformal motion of a set over a simply connected Hausdorff space can be extended to a quasiconformal motion of the whole sphere, over the same parameter space. I will also discuss differentiable motions and guiding quasiconformal motions in this talk.

Nov 22: Diogo Pinheiro (Brooklyn College)
A Renormalization Scheme for the Focal Decompositions of a Family of Mechanical Systems

We study the dynamics of a family of mechanical systems which includes the pendulum, on small neighborhoods
of an elliptic equilibrium and for long intervals of time. We characterize the dynamical behavior of such family through a
renormalization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the
asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of
mechanical systems. As a consequence we obtain a universal asymptotic focal decomposition for this family of mechanical
systems. Furthermore, we obtain that the asymptotic trajectories have a Hamiltonian character and compute the action of
each element in this family of trajectories. This is the first step towards the computation of the semiclassical expansions for
the quantum-mechanical propagators associated with this family of mechanical systems in the limit of small spatial deviation
and long time separation of the endpoints. This is joint work with C.A.A. de Carvalho, M.M. Peixoto and A.A. Pinto.

Dec 6: David Dumas (University of Illinois at Chicago/ICERM)
Computing the Image of Thurston's Skinning Map

Thurston's skinning map is a holomorphic map between Teichmuller spaces that arises in the construction of hyperbolic structures on compact 3-manifolds. I will describe the theory and implementation of a computer program that computes the images of skinning maps in some cases where the Teichmuller space has complex dimension one. The key to the method is that each point in the image of the skinning map represents an intersection between two Lagrangian subvarieties of the $SL(2,{\mathbb C})$ character variety of a surface group. The skinning image is computed by tracking the movement of these intersections as one of the varieties (the Bers slice) is changed. This is joint work with Richard Kent.